Induced charge in image problem

[Kaguro]

Homework Statement

Find induced charge density in the classic image problem.

Sorry, I uploaded my work in form of an image instead of typing.

I have tried to use polar coordinates to solve this. But i must have done something wrong. Please help me find the mistake.

 

[kuruman]

I do not believe you can use the dipole approximation here. It is valid for $r>>d$. Your potential does not go to zero on the plane which is the boundary condition in this problem.

 

[Kaguro]

I do not believe you can use the dipole approximation here. It is valid for $r>>d$. Your potential does not go to zero on the plane which is the boundary condition in this problem.

But my potential does go to 0 on the plane as theta is 90 then and so cos theta is 0.

 

[kuruman]

The entire plane is an equipotential therefore the expression must evaluate to zero for any value of angle $\theta$ and must also be zero at the origin. You should express the potential as the sum of the potentials from the point charge and its image. You would be wise to use cylindrical coordinates.